Facing a Big Decision? Math is Your Friend.

Many people, if not most, can make these decisions by trusting their gut.

If that’s you, well, I envy you. That doesn’t work for me. I will be waking up in the night in a panic with my heart pounding, wondering if I’m doing the right thing.

I trusted my gut when I bought my first (and only) house after deciding “It just feels right.” For the next year, I was convinced that when I left for a trip the whole thing would collapse in my absence.

After that, I realized I needed to base my decisions on some reason, weighing the cost against the benefits.

For some things it’s a no-brainer. The benefits of having a car here in Southern California far outweigh the cost, so we didn’t think too much about that one. But what about something that’s no so clear cut? I know I keep blabbing on about solar, but it’s a good example of an expense where the costs are high and benefits are less certain.

It’s time to break out the math. But stay with me. Thinking deeply about money in our daily lives can provide enormous insight into the world around us.

So, we want to be able to compare at the cost of the solar system to what we would be saving in electricity. Here’s what we spent on electricity in the last year:

One way to look at this is to figure out how many years of electricity bills will it take to pay for my solar system. My solar system will cost $14,700 after taxes. I spent $1241 on electricity last year. So my payback will be:

Net cost of solar system/annual electricity bill=12 years

If you want to stop there, fine. But if this strikes you as too simple, good for you! It doesn’t account for change over time. My solar system is supposed to last 30 years and a lot of things will change in that period.

For starters, my utility bill isn’t going to stay the same over the next 30 years. Not a chance! But how much will it rise? The Energy Information Association estimates that residential power bills will rise 2.5 percent a year through 2040. Applying that, my power bill will steadily rise until 2045 when it will be $2,540.

So that $2,540 is money I’m not spending since I’ve gone solar. I can spend it on whatever I want (or save it). So my solar system has freed up cash for my household for the next 30 years. If I were a business, I would call this positive cash flow.

Utility rates aren’t the only thing changing over time. What also will be changing is the value of the dollar. Inflation has been quiet of late but over the next three decades, it will rise again. In fact, there’s a chance it may come roaring back. In 30 years, a dollar will be worth less than a dollar today.

So how do we deal with all this change over time? The answer is a concept known as net present value (NPV). This is a very useful concept because once you understand it, you can use NPV to analyze all sorts of investments like stocks or bonds.

Here is a video that explains the concept in three minutes:

The rule is if your NPV is less than zero, it’s a bad deal. A net present value that is positive means that my solar project or any investment will save me more money than it costs.

It takes a bit of work to figure out net present value. You need to know how to use a spreadsheet, but it gives a much more accurate insight into whether an investment is worthwhile. When I’m spending $21,000 that matters!

Net present value accounts for the changing value of money over time. A dollar is worth slightly less next year than a dollar today. But how how much less? To calculate this, we need a rate of return. I used the 30-year Treasury bond rate, which as of today is 2.8 percent.

NPV is calculating interest in reverse. Let’s start with the interest calculation: If we bought a 30-year bond for $1000 today, next year it would be worth $1000*1.028 or $1,028.

Let’s look at it backwards. How much is a 30-year Treasury selling for $1,028 next year worth today? The answer is $1,028/1.028 or $1,000. Can you guess the NPV of the same bond selling for $1,056 in 2018? It’s $1,028/(1.028*1.028) or $1,000.

Excel and Google spreadsheets have an NPV function that makes calculating this easy. I applied this to my column of electricity bills growing 2.5 percent a year and subtract the cost of my solar and I get a net present value of $19,705. That’s the amount my solar system is saving me over the next 30 years! That’s way above zero, which means it’s a good deal.

There’s still one more way of looking at my solar system. It’s called Internal Rate of Return (IRR). Mathematically, it’s very close to NPV. We assume the NPV is zero and then solve for the rate of return. The IRR for my solar project is 11 percent. An investment that returns 11 percent a year? I’ll take it!

Both net present value and internal rate of return are very powerful tools that allow you to cut through a lot of B.S. Someone offers you an annuity that generates $2,000 a year, guaranteed. Is that a good deal? Renting a house and deciding whether to buy one? Now you can decide with some greater precision.

In fact, net present value is the secret weapon of stock guru Seth Klarman who praises it as a powerful analysis tool in his famous (in Wall Street circles) book Margin of Safety. Klarman writes, “When future cash flows are reasonably predictable and an appropriate discount rate can be chosen, NPV analysis is one of the most accurate and precise methods of valuation.”

How can this be used to pick stocks? We will cover the applications of NPV for stock analysis in our next a later post.